Off-diagonal holomorphic extension of real analytic functions on $\mathbb{C}^n \times\mathbb{C}^n$

Question

I am struggling trying to understand an statement in a paper I am reading:

Let $M$ be a complex manifold of dimension $2n$. Let's consider a function $\xi$: $M$ $\rightarrow$ $\mathbb{C}$ whose real and imaginary parts are real analytic functions. Let $diag$($M$,$M$) be the diagonal of $M\times M$. Then there is a holomorphic extension $\Xi$ of $\xi$ where $\Xi$ is defined in a open neighbourhood of $diag$($M$,$M$).

I already checked some references about real analytic functions but I could not find anything useful about holomorphic extensions of real analytic function in several complex variables. Even in the case $M$ = $\Omega$ a domain of $\mathbb{C}^n$ I could not figure out how to extend $\xi$($z_1$,.....,$z_n$) to a holomorphic function with the double of complex variables $\Xi$($z_1$,..,$z_n$,$\lambda_1$,..,$\lambda_n$) when we are off-diagonal.

Thanks

Answer 1

The statement you require (IIUC a simple consequence of analytic continuation) is e.g. in Bourbaki, Variétés différentielles et analytiques, Fascicule de résultats, 5.14.7, page 60. It requires the manifold to be paracompact.